We show global uniqueness in the fractional Calderón problem with a single measurement and with data on arbitrary, possibly disjoint subsets of the exterior. The previous work [10] considered the case of infinitely many measurements. The method is again based on the strong uniqueness properties for the fractional equation, this time combined with a unique continuation principle from sets of measure zero. We also give a constructive procedure for determining an unknown potential from a single exterior measurement, based on constructive versions of the unique continuation result that involve different regularization schemes.